Theorem: Full-code constant-complement criterion

Let $\mathcal{N}:L(H)\to L(Q)$ be CPTP, with Kraus operators $E_i:H\to Q$. The following are equivalent:

1. a CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_H$;

2. every ${E_j}^{†}{E}_i$ is a scalar multiple of $I_H$;

3. a complete complementary channel satisfies

   $${\mathcal{N}}^c(X)=\mathrm{Tr}(X)\tau$$

   for a fixed state $\tau$.

Proof. The Kraus complementary channel has entries

Equality on every matrix unit proves equivalence of (ii) and (iii). The scalar matrix is positive and has trace one by complete positivity and trace preservation.

Suppose (i), and let $K_a$ be Kraus operators of $\mathcal{D}$. The operators $K_a{E}_i$ form a Kraus family for the identity channel. The identity Choi matrix has rank one. Each positive rank-one Kraus contribution must lie in its one-dimensional range, so $K_a{E}_i=c_{ai}{I}_H$. Trace preservation gives

Conversely, diagonalize the positive scalar matrix by a unitary change of Kraus family and omit zero Kraus operators. Then

Their ranges are orthogonal. Put $P={\sum }_i{W}_i{W}_i^{†}$. For any fixed state ${\sigma }_H$, define

This map is completely positive and trace preserving. Substitution gives $\mathcal{DN}(X)={\sum }_i{\lambda }_{i}X=X$.